Limits of $\alpha$-harmonic maps

Critical points of approximations of the Dirichlet energy \`{a} la Sacks-Uhlenbeck
are known to converge to harmonic maps in a suitable sense.
However, we show that not every harmonic map can be approximated
by critical points of such perturbed energies. Indeed, we prove that
constant maps and the rotations of $S^2$ are the only critical points of $\E_{\alpha}$
for maps from $S^2$ to $S^2$ whose $\alpha$-energy lies below some threshold.
In particular, nontrivial dilations (which are harmonic)
cannot arise as strong limits of $\alpha$-harmonic maps.